Worksheet 1

Section 1.1

• Are all triangular systems echelon? If not, write down a triangular system that is not echelon.

Yes.

• Are all echelon systems triangular? If not, write down a echelon system that is not triangular.

No. Consider the system consisting of the single equation 0 = 0.

Section 1.2

• Find all solutions to the linear sytem of equations given by Example 3 in Section 1.2 in the book. If you’re tired of solving linear systems, just look up the answer in the book.

(x₁, x₂, x₃) = (2, 2, 0) + s₁( − 11,  − 4, 1).

• Explicitly write down 2 solutions. In other words, write down 2 points with actual numbers that solve the linear system.

(2, 2, 0), ( − 9,  − 2, 1).

• Find all solutions the homogenous linear system given by This is pretty much the same linear system but with the right hand side replaced by zeros.

(x₁, x₂, x₃) = s₁( − 11,  − 4, 1).

• Call the 2 solutions you found earlier a and b. Does a − b solve the homogenous linear system? The answer should be yes. Can you explain why?

Yes. See homegenous solutions section.